Summary: In this paper we determine the number of positive integer sequences \(a_1,a_2,\ldots,a_k\) such that \(1\leq a_1\leq \cdots \leq a_k \leq m\) and \(a_i\geq i\) for all \(i\in \{1,\ldots, k\}\). After that, we apply this result to calculate the number of anonymous, neutral and monotonic social welfare functions when only two alternatives are considered.